The quantum action principle

The analoguous behiaviour of the Poisson bracket and the commutator has been used to establishcorrespondence between classical and quantum mechanics. It is, however/shown in the next section following the derivation of Schwinger s quantum action principle that the correspondence goes deeper and that the analogous behaviour of the Poisson bracket and commutator is a consequence of the properties of infinitesimal canonical transformations which are common to both mechanics. 

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If the topological property which defines an atom is also one of physical significance, then it should be possible to obtain from quantum mechanics an equivalent mechanical definition. As demonstrated in Chapters 5 and 8, this can be accomplished through a generalization of the quantum action principle to obtain a statement of this principle which applies equally to the total system or to an atom within the system. The result is a single variational principle which defines the observables, their equations of motion, and their average values for the total system or for an atom within the system. 

The Lagrangian-based functional [i, 2] or 2] derives directly from the Lagrangian as employed in the quantum action principle. For a total system, both the Lagrangian- and Hamiltonian-based functionals yield identical variational results. This equivalence in variational behaviour is maintained for the corresponding subsystem functionals only if the subsystem is bounded by a zero-flux surface. Only an atomic region ensures an equivalence in both the values and the variational properties of the two types of functionals (eqns (E5.6a,b)) thereby preserving the properties obtained for a total system. 

If variations are effected in a quantum mechanical system, the corresponding change in the transformation function between the eigenstates 1, 1, ti> and q,2, t2> is i/k times the matrix element of the variation of the action integral connecting the two states. Equation (8.78) will be referred to as the quantum action principle. 

Srebrenik, S., Bader, R. F. W., Nguyen-Dang, T. T. (1978). Subspace quantum mechanics and the variational principle. J. Chem. Phys. 68, 3667-6679 Bader, R. F. W. Srebrenik, S., Nguyen-Dang, T. T. ibid.. Subspace quantum dynamics and the quantum action principle. 68, 3680-3691. 

The importance of Heisenberg s equation and of the corresponding hyper-vinal theorem for a stationary state in the description of the proMrties of the total system is maintained in the description of a subsystem. Indeed, the generalized action principle that is employed to establish the quantum... 

Schwinger s quantum action principle (1951) is developed in Section 8.2, and a statement of this principle, in which the action integral and Lagrange... 

This extension of quantum mechanics is indeed possible within the framework of the generalized action principle and, as stated in Section 1, its use has resulted in the identification of the proper open systems of quantum mechanics with the topological atom. The derivation of the quantum mechanics of a proper open system cannot be given here, but one can present an outline of the essential ideas. It is based on the reformulation of physics provided by the work of Feynman and Schwinger, an approach that makes possible the asking and answering of questions not possible within the traditional Hamiltonian framework. It is important that the reader appreciate that the alternative to the use of existing... 

The form of the action principle given above was first applied to quantum mechanics to describe the time evolution of pure states (i.e. 

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

On a more philosophical or meta-physical level, one may suspect that free will and consciousness may have some quantum mechanical origin rooted in the Heisenberg Uncertainty Principle. Perhaps at some neurological level an electron at a synapse exists in a superposition of two or more states that ultimately results in someone making some sort of decision. Should I run for President, or not Should I get married, or not . Perhaps there are two states with eigenvalues yes or no that asymptotically lead to very different actions. Does quantum theory enter into our decision making process Perhaps the brain itself acts as some sort of quantum computer taking... 

These results were obtained by using the time-dependent quantum mechanical evolution of a state vector. We have generalized these to non-equilibrium situations [16] with the given initial state in a thermodynamic equilibrium state. This theory employs the density matrix which obeys the von Neumann equation. To incorporate the thermodynamic initial condition along with the von Neumann equation, it is advantageous to go to Liouville (L) space instead of the Hilbert (H) space in which DFT is formulated. This L-space quantum theory was developed by Umezawa over the last 25 years. We have adopted this theory to set up a new action principle which leads to the von Neumann equation. Appropriate variants of the theorems above are deduced in this framework. 

At the risk of being redundant, we may state here the salient features of the TD-functional formalism. The first requirement is a variational principle, and for a time-dependent quantum description only a stationary action principle is available. With this a mapping theorem is established which turns the action functional into a functional of relevant physical quantities (which are the expectation values), and the condition of stationarity is now in terms of these variables instead of the entire density matrix. Thus the stationary property with respect to the density matrix now becomes one with respect to all the variables... 

In this appendix, we establish the important result that the TD stationary action principle used in the derivation of the TD-functional theory [Eq. (54)] is the same as the stationary principle for the effective action functional of Baym [Bq. (89)]. Thus the action functionals appearing in the two formulations are different representations of the same quantum action leading to two different optimization strategies. We give a version of the work of Jackiw and Kerman [54] (hereafter referred to JK) on this subject, adapted here to LQD. Consequences of this important result in the development of the functional theory presented here are given. 

These rules are a part of the formalism of quantum mechanics that is recovered from Schwinger s action principle. 

We are now in a position to perform a generalized variation of the action integral for an open system to demonstrate that Schwinger s principle of stationary action can be extended in such a manner as to provide a quantum definition of an atom in a molecule. We shall be considering the change in the atomic action integral 2] of eqn (8.111) ensuing from variations... 

One recognizes the first term in eqn (8.136) as the variation in the quantum mechanical current density (eqns (5.94) and (5.95)). It is obtained by combining the surface term arising from the variation with respect to VT with the surface term arising from the imposition of the variational constraint, eqn (8.135). Thus the variation of the surface of the subsystem together with the restriction that the subsystem be an atom bounded by a zero-flux surface causes the quantum mechanical current density to appear in the variation of the action integral, a term whose presence is a necessary requirement for the proper description of the properties of an open system. It is now demonstrated that eqn (8.136) is the atomic equivalent of the principle of stationary action. 

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